First Steps into Advanced Mathematics

Professor: Erika L.C. King

Email: eking@hws.edu

Office: Lansing 304

Phone: (315) 781-3355

Home Page

Office Hours: M: 10:00-11:30am, T: 9:45-11:15am, W: 2:45-3:45pm, Th: 4:00-5:00pm, and by appointment

Class Schedule: held TTh 2:00-3:30pm in Eaton 110

Course Syllabus

Proof Writing and Presentation Tips

Individual Assignments Policy

Course Grade Scale

**During Spring Break I will be working to determine how best to organize our class for remote learning. I encourage you to read your emails closely, and to
contact me with any questions, concerns or suggestions you have. If you have issues with internet accessibility, please let me know what those are. While this will
not be ideal, we will work together to make the best learning experience we can.**

**Look for an email from me no later than Monday, March 23rd detailing our strategy
for proceeding. At this point, you should plan to be available for class at the usual time on Tuesday, March 24th (2:00pm EST), perhaps via Zoom, but again, I will
determine that in the next week. **

**I may contact you via email in the middle of spring break to ask questions about some options and will appreciate responses as
soon as it is possible for you to give them. While it will be helpful for all of us to stay on track with material and assignments, we will need to be a bit more
flexible with some deadlines. Be in contact with me about your needs as the semester progresses. **

**For this next week, you do not need to work on material for
this course (although it may be fun and a good distraction for you!); concentrate more on planning for how we will work for the rest of the semester and more
importantly on your health (physical and emotional) and your family. Again, I encourage you to come to me with any questions, concerns or suggestions you have.
Thank you for your patience with me as I learn new tools like Canvas and Zoom. Also thank you for being a great class; that gives me hope that we can still have
a good rest of the semester in whatever form(s) it takes!**

**Quiz 6 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, March 5th and Tuesday, March 10th.**

**Journal Homework for class Tuesday, March 10:**

- Thank you for being so diligent and enthusiastic on Thursday! It was wonderful to come into class and see all of you working so hard and asking good questions. Keep up the great work!
- Review our work in Thursday's class on graphs, complements and isomorphisms. Make sure the problems in Section 3.8 (pages 78-79) make sense. Check to see if we answered your question you put in your journal for Thursday, if not plan to come to an office hour or ask it in class!
- Play the Dot Game on six vertices! Or try coloring the edges of a complete graph on six vertices without creating a triangle in a single color. Can you do it? That is, is it possible to have a draw in the game on six vertices? If so, show it! If not, try to prove why not! Put your results in your journal and be ready to discuss them and/or show your trials and conjectures on your quiz!
- Start reading Chapter 4. In particular, read pages 91-94. This is an introduction to a new type of proof! Cool stuff!
- Complete problems 1, 2, 8 and 12 in Section 3.13 (pages 85-86).
- Presentation Opportunity! Part 1: Draw a graph with degree sequence (1, 1, 2, 2, 2, 3, 3). Part 2: Carefully prove that there exists a graph having degree sequence (3, 3, 4, 4, 4, 5, 5) without actually drawing such a graph. (Hint: How does Part 1 help you?) Remember you are welcome (and highly encouraged!) to discuss it with me before class!
- Outline of work for Tuesday's journal:
- Dot Game conjectures and results on six vertices
- Notes from reading Chapter 4 pages 91-94
- Section 3.13: 1, 2, 8, 12
- Your question on the material.

**On Wednesday, March 11th at 5:00pm in Napier 201, there will be a colloquium given by Catalina Garcia Tomas, Kaitlyn Geraghty, Connor Parrow, and Yifei Tao
about their trip to the Nebraska Conference for Undergraduate Women in Mathematics. Refreshments will be served at 4:45pm.
I hope you can make it!**

**Journal Homework for class Thursday, March 12:**

- Review our work reviewing one-to-one and onto functions, discussing the Dot Game, and proving that every tree on at least two vertices has at least one leaf from Tuesday's class. Write down any questions you have on that material or material from Chapter 3 in general so that we can address them in Thursday's class. Check to see if we answered your question you put in your journal for Tuesday, if not plan to come to an office hour or ask it in class!
- Take Two (I didn't get to this in class on Tuesday): Presentation Opportunity! Part 1: Draw a graph with degree sequence (1, 1, 2, 2, 2, 3, 3). Part 2: Carefully prove that there exists a graph having degree sequence (3, 3, 4, 4, 4, 5, 5) without actually drawing such a graph. (Hint: How does Part 1 help you?) Remember you are welcome (and highly encouraged!) to discuss it with me before class!
- Read Sections 3.9-3.11 in the text. (This covers pages 80-84.) In Section 3.9 Prof. belcastro reviews the secret to the Dot Game we discussed in class. See if you can explain to a friend why there will never be a tie in the Dot Game if you perform the game on six or more vertices.
- Complete the Check Yourself problems at the end of Section 3.9 (page 81).
- Complete problems 4, 5 and 6 in Section 3.13 (page 86).
- Read/reread Sections 4.1-4.2 in the text. (This covers pages 91-99.)
- Complete the Check Yourself problems at the end of Section 4.2 (page 100).
- Outline of work for Thursday's journal:
- Notes from Sections 3.9-3.11 including definitions
- Section 3.9 Check Yourself Problems
- Section 3.13: 4, 5, 6
- Notes from Sections 4.1-4.2 including definitions
- Section 4.2 Check Yourself Problems
- Your question on the material.

**Collected Homework (Due Friday, March 13 at 3:00pm):**

- Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
- Do NOT staple the Collaborative and Individual assignments together. However, you SHOULD staple the separate assignments if they have more than one page.
- Collaborative Assignment:
- Remember for this part of the assignment you ARE allowed to discuss it with your classmates or other faculty or staff; HOWEVER, you may not use internet resources and your write up must still be your own.
- You MAY complete this assignment in LaTeX, but it is not required.
- (1) Suppose a graph contains eight vertices and 21 edges. Explain how you know that this graph must contain at least one vertex that has degree less than or equal to 5 and at least one vertex that has degree greater than or equal to 6.
- (2) Degree Sequences Exercise: Determine if the following are degree sequences of some
**simple**graph G. If a sequence is a degree sequence, give an example of a graph G that has that degree sequence. If it is not a degree sequence, explain why it cannot be one (note that "I couldn't make a graph with that sequence" is not an explanation).- (a) (1, 1, 2, 2, 3, 4, 7)
- (b) (1, 1, 2, 2, 3, 4, 4, 5)
- (c) (1, 2, 2, 2, 3, 3, 3, 4, 5)

- (3) Another Degree Sequence Exercise: Find two non-isomorphic graphs with the degree sequence (1, 1, 1, 1, 2, 2, 3, 3). Draw them and then prove why they are non-isomorphic.
- (4) Complete Problem 11 in Section 3.13 (page 86). Your answers to the last two parts of this question should be proofs!
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proofs great!
- Individual Assignment:
- Remember for this part of the assignment you are NOT allowed to discuss it with your classmates and others; you may ONLY discuss it with me. You may NOT use internet resources.
- Remember this part of the assignment MUST be typed in LaTeX.
- The policy for Individual Assignments is summarized here.
- Complete Problem 13 in Section 3.13 (page 87). Your solution should be a rigorous proof!
- Complete Problem 21 in Section 3.13 (page 87). Your solution should be a rigorous proof!
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great!
- Resubmissions:
- You may also resubmit the Individual Assignments that were due February 14 (Final Chance for this one!), February 21 (Final Chance: March 27), February 28 (Final Chance: March 27). Remember that you must turn in your earlier draft(s) with any resubmission. Also note that you may turn in resubmissions on any day of the week, you need not wait until a Friday. Review our Individual Assignment Policy and be sure that you do at least one resubmission for each Individual Assignment!

**Quiz 5 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, February 20th, Thursday, February 27th and
Tuesday, March 2nd.**

**Journal Homework for class Tuesday, March 3:**

- Review our work in Thursday's class on functions, the properties of one-to-one and onto, and graphs. Check to see if we answered your question you put in your journal for Thursday, if not plan to come to an office hour or ask it in class!
- Work through the second page of the group worksheet that we were doing in class and be ready to share the results with your group. Read through the third page of the handout and write down a few notes about how you would prove the given statements.
- Reread Section 3.3.2, pages 63-65, in the text. Finish working through questions 1-4 on pages 64-65.
- Read Section 3.3.3 pages 65-66, and play the Dot Game with your friends and/or relatives. Try to answer the questions on page 65 about the Dot Game. After you have played and thought about the Dot Game for a few days, read the hints on page 66. Does this influence your conclusions?
- Read Sections 3.4 in the text. (This covers page 66 and the very beginning of page 67.) Do NOT read Section 3.5. Section 3.4 confirms what we were discussing about the number of possible functions.
- Complete problem 7 in Section 3.13 (page 86) in your journal.
- Outline of work for Tuesday's journal:
- Notes from reading Sections 3.3.2 and 3.3.4 including definitions
- Section 3.3.2: 1-4
- Section 3.3.3 experiments
- Section 3.13: 7
- Your question on the material.

**Journal Homework for class Thursday, March 5:**

- Review our work on graphs from Tuesday's class. Write down any questions you have so that we can address them in Thursday's class. Check to see if we answered your question you put in your journal for Tuesday, if not plan to come to an office hour or ask it in class!
- Continue to play the Dot Game with your friends (especially on six vertices), and see if you can come up with any new conjectures, ideas to support your claims, or counterexamples to show your claims were false. All three are important mathematical tasks! Add these attempts to your work in your journal reflecting on the questions on page 65 about the Dot Game. We will discuss your results!
- Reread Section 3.5 in the text (the material you read in class). This covers pages 67-70. Be sure to add the vocabulary words to your list - there are lots!
- Make sure you understand all the Check Yourself problems at the end of Sections 3.5 that we were working on in class (pages 70-71). Remember that you can check answers in the back of the text as well. These need NOT go in your journal, but you should write down any questions you have about them.
- Read Sections 3.6-3.7 in the text. (This covers pages 71-77.)
- Complete the Check Yourself problems at the end of Sections 3.6 and 3.7 (pages 74 and 78 respectively).
- Complete problems 2 and 5 in Section 3.8 (pages 78-79).
- Presentation Opportunity! I will looking for someone to discuss isomorphisms! The presenter would introduce the definition of isomorphism and illustrate the definition by presenting questions 2 and 5 from Section 3.8. I would like the presenter to speak with me, preferably in office hours, but over email might be possible as well, about the presentation before class! Please be in touch if you want to take advantage of this opportunity.
- Look for an email from me about class on Thursday. Please be there at 2pm. I may be there then or I will arrive by 2:15. I will send you information about how to begin class if I am not there right at 2pm.
- Outline of work for Thursday's journal:
- Conjectures and ideas about the Dot Game
- Notes from Section 3.5 including definitions and theorems
- Notes from Section 3.6 including definitions
- Section 3.6 Check Yourself Problems
- Notes from Section 3.7 including definitions
- Section 3.7 Check Yourself Problems
- Section 3.8: Problems 2 and 5
- Your question on the material.

**On Wednesday, March 11th at 5:00pm in Napier 201, there will be a colloquium given by Catalina Garcia Tomas, Kaitlyn Geraghty, Connor Parrow, and Yifei Tao
about their trip to the Nebraska Conference for Undergraduate Women in Mathematics. Refreshments will be served at 4:45pm.
I hope you can make it!**

**Collected Homework (Due Friday, March 6 at 3:00pm):**

- Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
- Do NOT staple the Collaborative and Individual assignments together. However, you SHOULD staple the separate assignments if they have more than one page.
- Collaborative Assignment:
- Remember for this part of the assignment you ARE allowed to discuss it with your classmates or other faculty or staff; HOWEVER, you may not use internet resources and your write up must still be your own.
- You MAY complete this assignment in LaTeX, but it is not required.
- Complete these THREE questions about determining whether a relation is a function, one-to-one or onto.
- I chose not to add any new questions to the assignment due on Friday.
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proofs great!
- Individual Assignment:
- Remember for this part of the assignment you are NOT allowed to discuss it with your classmates and others; you may ONLY discuss it with me. You may NOT use internet resources.
- Remember this part of the assignment MUST be typed in LaTeX.
- The policy for Individual Assignments is summarized here.
- Prove these statements about composition and one-to-one functions.
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great!
- Resubmissions:
- You may also resubmit the Individual Assignments that were due February 7 (Final Chance for this one!), February 14 (Final Chance: March 13), February 21 (Final Chance: March 27), February 28 (Final Chance: March 27). Remember that you must turn in your earlier draft(s) with any resubmission. Also note that you may turn in resubmissions on any day of the week, you need not wait until a Friday. Review our Individual Assignment Policy and be sure that you do at least one resubmission for each Individual Assignment!

**Review Session: Monday, February 24th 4:30-5:30pm! The review session will be held in Lansing 301, the
seminar room on the third floor of Lansing. **

**Journal Homework for class Tuesday, February 25:**

- Exam Day!
- Prepare for your exam! Read the review sheet here. Have confidence in your abilities!
- Remember to bring in your picture if you forgot to bring it to your appointment! Last chance for full credit on that first assignment on Thursday!

**Note that there will be shorter collected assignment due Friday that will be posted on Tuesday.**

**Journal Homework for class Thursday, February 27:**

- Review our work on functions from last Thursday's class. Write down any questions you have so that we can address them in Thursday's class. Check to see if we answered your question you put in your journal for last Thursday, if not plan to come to an office hour or ask it in class!
- Reread Sections 3.1-3.2 in the text. (This covers pages 57-62.)
- Finish working through the Practice with Functions worksheet that we were working on in class on Thursday, February 20. Be ready to share your results!
- Do the questions listed in Section 3.3.1 on page 63 in your text. The first two questions are like the questions we were doing on the Practice with Functions worksheet, but with function notation instead of diagrams. Remember to try to justify your answers!
- Outline of work for Thursday's journal:
- Add any additional notes you have on the Section 3.1-3.2 reading
- Solutions to 3.3.1 questions
- Your question on the material.

**On Thursday, February 27th at 4:30pm in Napier 201, there will be a colloquium given by William Smith alum Avery Wickersham '19. She will be telling us about
her experience studying abroad in the Budapest Semesters in Mathematics Education program. Refreshments will be served at 4:15pm. I hope you can make it!**

**Because of the colloquium, my office hours on Thursday will be as soon as I get back to my office (likely before the usual 3:45pm start time) from our class
until about 4:20pm. If you cannot make that time and have questions, please be sure to contact me to make alternate arrangements.**

**Collected Homework (Due Friday, February 28 at 3:00pm):**

- Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
- Do NOT staple the Collaborative and Individual assignments together. However, you SHOULD staple the separate assignments if they have more than one page.
- Collaborative Assignment:
- Remember for this part of the assignment you ARE allowed to discuss it with your classmates or other faculty or staff; HOWEVER, you may not use internet resources and your write up must still be your own.
- You MAY complete this assignment in LaTeX, but it is not required.
- Prove the following: For all real numbers x and y, if x+y is irrational, then x is irrational or y is irrational. Think carefully about which method of proof you should use.
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proofs great!
- Individual Assignment:
- Remember for this part of the assignment you are NOT allowed to discuss it with your classmates and others; you may ONLY discuss it with me. You may NOT use internet resources.
- Remember this part of the assignment MUST be typed in LaTeX.
- The policy for Individual Assignments is summarized here.
- Prove the statement found here.
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great!
- Resubmissions:
- You may also resubmit the Individual Assignments that were due January 31 (Final Chance for this one!), February 7 (Final Chance: March 6), February 14 (Final Chance: March 13), and February 21 (Final Chance: March 27). Remember that you must turn in your earlier draft(s) with any resubmission. Also note that you may turn in resubmissions on any day of the week, you need not wait until a Friday. Review our Individual Assignment Policy and be sure that you do at least one resubmission for each Individual Assignment!

**Quiz 4 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, February 13th and Tuesday, February 18th.**

**Remember our first exam is Tuesday, February 25, in class. Work on making yourself a review sheet that you can compare with the one I give you at the
end of this week. Think about whether you would like a review session on Sunday, February 23rd or Monday, February 24th. **

**Journal Homework for class Tuesday, February 18:**

- Review our work in Thursday's class on types of proof, especially proof by contradiction, power sets, and proving set equality via element arguments. Check to see if we answered your question you put in your journal for Thursday, if not plan to come to an office hour or ask it in class!
- Work through the second page of the group worksheet that we were doing in class and be ready to share the results with your group. Read through the third page of the handout and write down a few notes about how you would prove the given statements.
- On page 30 of our text when Professor belcastro is defining union, intersections, and so on, she shows how we would notate taking unions and intersections of many sets, including an infinite number of sets. These are often called Indexed Sets (they are organized by their indices!). Review page 30 and then read Section 1.8 in Hammack's book on Indexed Sets (pages 25-28).
- Complete numbers 4 (d-f) and 7 in Section 2.4 (page 45).
- Complete problems 10, 15 and 18 in Section 2.9 (page 54) of our text.
- Outline of work for Tuesday's journal:
- Notes from reading Section 1.8 in Hammack's text
- Section 2.4: 4 (parts d-f), 7
- Section 2.9: 10, 15, 18
- Your question on the material.

**Journal Homework for class Thursday, February 20:**

- Review our work in Tuesday's class. Write down any questions you have so that we can address them in Thursday's class. Check to see if we answered your question you put in your journal for Tuesday, if not plan to come to an office hour or ask it in class!
- Finish working through the group work handout on Indexing Sets. Be prepared to share your ideas and questions.
- Finish working through Part III of the group work handout on Equal Sets and Power Sets. Be prepared to share your ideas and questions.
- Presentation Opportunity! Present number (2) of Part III from the Equal Sets and Power Sets worksheet. This should be an element argument, but you may also use theorems that we have previously proved if that is helpful! Remember you are welcome (and highly encouraged!) to discuss it with me before class! You should come to class prepared and you are welcome to use your notes.
- Read Sections 3.1-3.2 in the text. (This covers pages 57-62.)
- Complete the Check Yourself problems at the end of Section 3.2 (pages 62-63).
- Outline of work for Thursday's journal:
- Notes from reading Sections 3.1-3.2, including definitions
- Section 3.2, Check Yourself problems
- Your question on the material.

**Collected Homework (Due Friday, February 21 at 3:00pm):**

- Collaborative Assignment:
- You MAY complete this assignment in LaTeX, but it is not required.
- 1. Consider the following theorem: For all integers x, if 4 does not divide x^2, then x is odd.
- (a) State the assumptions and final conclusion if proving this statement by direct proof.
- (b) State the assumptions and final conclusion if proving this statement by contraposition.
- (c) State the assumptions and final conclusion if proving this statement by contradiction.
- (d) Choose one of the methods above and use it to write a proof of the theorem.

- 2. (a) Use proof by contradiction to prove: For all integers a and b, if ab is an even number, then either a or b is an even number. (b) Explain why proof by contradiction was a good method to use. (Note: recall just because contradiction is a GOOD method, does not mean that it is the ONLY method.)
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proofs great! Especially review tips 4, 5, 6, 11, and 13!
- Individual Assignment:
- Remember this part of the assignment MUST be typed in LaTeX.
- The policy for Individual Assignments is summarized here.
- Complete problem 19 in Section 2.9 (page 54). Note that you do not need to use the same proof method for both directions!
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 13!
- Resubmissions:
- You may also resubmit the Individual Assignments that were due January 31 (Final Chance: February 28), February 7 (Final Chance: March 6) and February 14 (Final Chance: March 13). Remember that you must turn in your earlier draft(s) with any resubmission. Also note that you may turn in resubmissions on any day of the week, you need not wait until a Friday. Review our Individual Assignment Policy and be sure that you do at least one resubmission for each Individual Assignment!

**Quiz 3 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, February 6th and Tuesday, February 11th.**

**Journal Homework for class Tuesday, February 11:**

- Review our work in Thursday's class on logic. Check to see if we answered your question you put in your journal for Thursday, if not plan to come to an office hour or ask it in class!
- Finish the group worksheet that we were doing at the end of class and be ready to share the results with your group.
- Review power sets from page 28 in the text and our groupwork sheets in class. We will do another worksheet on these as well, so be sure you have reviewed the definitions, reworked some examples, and write down any questions you have about power sets.
- Complete numbers 1, 3 and 5 in Section 2.4 (pages 44-45).
- Complete problem 5 in Section 2.9 (page 53) of our text.
- Read Section 2.5 in the text. (This covers pages 46-47.) This section introduces two new types of proof: proof by contraposition and proof by contradiction! Woo-hoo!
- Complete the Check Yourself problems at the end of Section 2.5 (page 48).
- In your journal, write down the two main steps for a proof by contraposition, then do the same for contradiction. (Hint: the two steps should start with "Assume" and "Show"!)
- Our First Presentation Opportunity! Consider the symbolic statement shown here. Negate this statement. Then illustrate a truth table for the negation. You will need to both explain your work and write it on the board. Please feel free to come and discuss this with me before Tuesday's class!
- Outline of work for Tuesday's journal:
- Section 2.4: 1, 3, 5
- Section 2.9: 5
- Notes from reading Section 2.5, including definitions
- Section 2.5, Check Yourself problems 1, 2 and 3
- Two main steps for each of proof by contraposition and proof by contradiction
- Your question on the material.

**The results of our survey are in! Our midterms will take place during regular class time on the dates listed in the syllabus. There will be optional review
sessions scheduled the Sunday or Monday before each exam. The syllabus on this website (link above) has been updated to reflect this.**

**Journal Homework for class Thursday, February 13:**

- Review our work in Tuesday's class. Write down any questions you have so that we can address them in Thursday's class. Check to see if we answered your question you put in your journal for Tuesday, if not plan to come to an office hour or ask it in class!
- Finish working through the group work handout on Types of Proof. Be prepared to share your ideas and questions.
- Another Presentation Opportunity! Present part (g) of the Types of Proof worksheet from Tuesday's class. This is the proof by contraposition. Remember you are welcome (and highly encouraged!) to discuss it with me before class! You should come to class prepared and you are welcome to use your notes.
- Read Section 2.3 of Richard Hammack's chapter on Logic (this is on pages 42-45 of his text). Pay special attention to the different forms of "if-then" on page 44. In your journal, write all ten ways of expressing the statement: If a sequence is bounded and monotonic, then the sequence is convergent. (That is, this one and nine more ways!) Think about whether it makes sense to you that these are all equivalent.
- Complete problems 4(a)-(c) (Note that you have actually already done most of (a)! Also the word "Prove" in part (c) should really be something like "Illustrate".), 8 and 9 in Section 2.4 (page 45) in your journal.
- Read Section 2.7 in the text. (This covers pages 49-50.) Note that part of this section asks you to reread some other material! Do that too! When she says "reread Section 3", she means of the Preface back on pages xxxiii and xxxiv.
- Outline of work for Thursday's journal:
- Notes from Section 2.3 of Hammack's book
- The ten equivalent ways of writing "If a sequence is bounded and monotonic, then it is convergent."
- Section 2.4: 4 (parts a-c), 8, 9
- Notes from reading Section 2.7 in our duck book
- Your question on the material.

**Collected Homework (Due Friday, February 14 (Happy Valentine's Day!) at 3:00pm):**

- Collaborative Assignment:
- You MAY complete this assignment in LaTeX, but it is not required.
- Negate the following statements. It may be helpful to rewrite the statement in
an equivalent form (using if-then, there exists, etc.) before negating it. Make your statements positive where possible. You may, but do not need to, show
me a translation of the original sentence into symbols.
- All HWS students prefer Oba Express over Ichiro.
- If there is a largest natural number, then it is 1.
- Some children like to play basketball.
- For all colors k, there exists a textbook b such that, if b is k, then Philipp is happy and Tyler is sad.
- If G is a bipartite graph, then G has no odd cycles.
- If x is divisible by 10, then x is divisible by 2 and x is divisible by 5.
- There exists a graph G such that G is Hamiltonian and G contains a cut-vertex.
- For all graphs G, if G is Eulerian, then every vertex has even degree.

- Complete problems 8 and 10 from Section 2.5 of Hammack's text. This is on page 50 in the chapter on Logic. For question 10, DO show a truth table and then use sentences to explain your answer. Note that Hammack says it can be done without a truth table (which is true!), but I would like you to show me the truth table. In your truth tables, make sure the difference between your T's and F's is very clear. Any ambiguous letters will be counted wrong.
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proofs great! Especially review tips 4, 5, 6, 11, and 13!
- Individual Assignment:
- Remember this part of the assignment MUST be typed in LaTeX.
- The policy for Individual Assignments is summarized here.
- Complete the question found here.
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 13!
- Resubmissions:
- You may also resubmit the Individual Assignments that were due January 31 (Final Chance: February 28) and February 7 (Final Chance: March 6). Remember that you must turn in your earlier draft(s) with any resubmission. Also note that you may turn in resubmissions on any day of the week, you need not wait until a Friday. Review our Individual Assignment Policy and be sure that you do at least one resubmission for each Individual Assignment!

**Due to needing to visit another professor's class, I need to adjust my Monday office hours. They will be 9:30am-10:45am this Monday, February 3rd. If you
have a question and cannot make this time, please email me.**

**Quiz 2 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, January 30th and Tuesday, February 4th.**

**Journal Homework for class Tuesday, February 4:**

- Review our work in Thursday's class on sets. Check to see if we answered your question you put in your journal for Thursday, if not plan to come to an office hour or ask it in class!
- In groupwork this week we will be looking at sets of real numbers. Hence we will be using interval notation. To review interval notation, check out the
book that Prof. belcastro recommends for "further reading", Richard Hammack's book,
__Book of Proof__. In particular, check out page 7 of his chapter on Sets. To review how we find intersections and unions of sets expressed in interval notation, check out page 18. This is not required for your journal, but it would be beneficial to you to write down anything that you need to refresh or that you have questions about. - Reread Section 2.2 and read Section 2.3 in the text. (This covers pages 26-44.)
- Complete Check Yourself problems 1, 4, 5, and 8 at the end of Section 2.3 (page 44).
- Read the Proof Writing and Presentation Tips website. This
may be revised from time to time. Use this as a reference when you are preparing your homework and presentations for class. Some of this will not mean anything to you
yet, but it will give you an idea of what I will be looking for as I grade your work. Note that
there is a link to the
Proof Writing and Presentation Tips website at the top of this website as well. You should refer to it regularly throughout the semester. - Outline of work for Tuesday's journal:
- Notes from reading Section 2.3, including definitions and laws
- Section 2.3, Check Yourself problems 1, 4, 5 and 8
- Your question on the material.

**Journal Homework for class Thursday, February 6:**

- Review our work in Tuesday's class. Make sure all parts of the group work make sense to you, and if not, write down any questions you have so that we can address them in Thursday's class. Check to see if we answered your question you put in your journal for Tuesday, if not plan to come to an office hour or ask it in class!
- Reread Sections 2.2 and 2.3 in the text. (This covers pages 26-44.) There is a lot of information here. Any new questions? Does it make more sense now that we worked on 2.2 in class?
- Complete Check Yourself problems 2, 3, 6 and 7 at the end of Section 2.3 (page 44).
- Recall that Prof. belcastro recommends checking out Richard Hammack's book,
__Book of Proof__, which is free online, as a supplement. Read (in full or at least some of) Sections 1.1-1.3 from his chapter on Sets. This covers pages 3-15, and should be a review of material we have discussed, though from a slightly different perspective. Then do the following exercises from his text: Section 1.1 numbers 12, 26 and 32 (this is on pages 7-8 of his text); Section 1.2 number 2 (this is on page 11 of his text); Section 1.3 number 8 (this is on page 15 of his text). - Outline of work for Thursday's journal:
- Make sure your notes from Sections 2.2 and 2.3 are complete.
- Section 2.3, Check Yourself problems 2, 3, 6 and 7
- Exercises from the Hammack book: Section 1.1: 12, 26 and 32; Section 1.2: 2; Section 1.3: 8
- Your question on the material.

**Collected Homework (Due Friday, February 7 at 3:00pm):**

- Collaborative Assignment:
- You MAY complete this assignment in LaTeX, but it is not required.
- Suppose m is an even number and n is an odd number. Is 3m+5n even or odd? Write a theorem that gives the answer to this question, then prove it. Be sure to state your theorem. Prove it from scratch using definitions.
- Complete problems 11, 13 and 14 in Section 2.9 (page 54). Note that both Problems 11 and 14 involve proofs. For Problem 13, be sure to show the step by step process, that is Venn diagrams illustrating each set needed like in Figure 2.9 on page 33. Also assume that A, B and C are subsets of some universal set (also like in Figure 2.9).
- Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proofs great! Especially review tips 4, 5, 6, 11, and 13!
- Individual Assignment:
- Remember this part of the assignment MUST be typed in LaTeX.
- The policy for Individual Assignments is summarized here.
- Prove this theorem. In other words, show that
union distributes over intersection
**using an element argument**. (Hints: Reread Section 2.2.3 on page 29 of our text. Recall that there are two parts to a proof that proves two sets are equal! Note that in Example 2.3.5, Prof. belcastro shows that intersection distributes over union. We are NOT proving this theorem that way. Here we are using an element argument. Also, I think Prof. belcastro makes an assumption not given in the text in her proof, so I claim the proof is incomplete anyway!) - Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 13!
- Resubmissions:
- You may also resubmit the Individual Assignment that was due January 31 (Final Chance: February 28). Remember that you must turn in your earlier draft(s) with any resubmission. Also note that you may turn in resubmissions on any day of the week, you need not wait until a Friday. Review our Individual Assignment Policy and be sure that you do at least one resubmission for each Individual Assignment!

**Quiz 1 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, January 23rd and Tuesday, January 28th.**

**Journal Homework for class Tuesday, January 28:**

- Review our work in Thursday's class and make any revisions to your journal you think necessary. In particular add the answer to your question for Thursday or plan to come to an office hour if you haven't figured the answer out yet! (Remember that you should write down at least one question on the material for each class day.)
- Read Sections 1.5-1.6 in the text. (This covers pages 14-20.)
- Complete the Check Yourself problems at the end of Section 1.5 (page 19).
- Complete problem 17 in Section 1.7 (page 21).
- On page 18, Prof. belcastro introduces
**existence**proofs/theorems that I referred to a couple of times in class on Thursday. What are and what makes them different from other proofs/theorems? Here is an existence theorem: "There exists an integer whose cube equals its square." Try writing a nice proof of this theorem in your journal. - Prepare for our first quiz! This mostly means that you should make sure your journal assignments are complete. Anything that was required in the journal for Thursday the 23rd and today is fair game for the quiz! Remember to organize your journal clearly. Labeling each problem clearly is important for the quiz. Make sure they are clear to you in case I refer to a specific section and problem number on the quiz.
- Outline of work for Tuesday's journal:
- Notes from reading Sections 1.5-1.6, including definitions and statements of theorems
- Section 1.5, Check Yourself problems
- Section 1.7, Problem 17
- Proof of: There exists an integer whose cube equals its square.
- Your question on the material.

**LaTeX Homework due Tuesday, January 28 by 5:00pm:**

Create a reproduction of the handout I gave you in class on Thursday with Overleaf. Email me your LaTeX file containing your work. This assignment is out of ten points and contributes to your overall homework grade.

**Journal Homework for class Thursday, January 30:**

- Review our work in Tuesday's class. Make sure the group presentations made sense to you, and if not, write down any questions you have so that we can address them in Thursday's class. Check to see if we answered your question you put in your journal for Tuesday, if not plan to come to an office hour or ask it in class!
- Read Sections 2.1-2.2 in the text. (This covers pages 25-34.)
- Complete Check Yourself problems 1-6 at the end of Section 2.2 (page 34).
- Remember to put at least one question about the reading (or other material) in your journal and be ready to ask it at the start of class!
- Complete problems 13 and 25 in Section 1.7 (page 21).
- Outline of work for Thursday's journal:
- Notes from reading Sections 2.1-2.2, including definitions and statements of theorems
- Section 2.2, Check Yourself problems
- Section 1.7, Problems 13 and 25
- Your question on the material.

**Collected Homework (Due Friday, January 31 at 3:00pm):**

- Collaborative Assignment:
- You MAY complete this assignment in LaTeX, but it is not required.
- Complete problems 10 and 12 in Section 1.7 (page 21).
- Complete problem 20 in Section 1.7 (page 22). Be sure to explain your answer.
- Complete this All Connected question: As of June, 2013 there were 327,577,529 cell phones in the US. How many must have the same last 7 digits in their phone numbers, i.e., *** - ****? Explain carefully referring to principles where appropriate.
- Individual Assignment:
- Remember this part of the assignment MUST be typed in LaTeX.
- The policy for Individual Assignments is summarized here.
- Complete problem 23 in Section 1.7 (page 22). Be sure to give details and explain why.
- Prove or disprove: There exist integers m and n such that 15m+12n=21.

**Welcome to First Steps into Advanced Mathematics!!!**

**Collected Homework (Due Wednesday, January 22 by 4:00pm):**

- Fill out this autobiographical questionnaire. Be sure to leave the top portion (above where you place your name) blank.
- Read the article "The Secret to Raising Smart Kids", by Carol Dweck.
- Write an essay as assigned on the syllabus. Hand both the questionnaire and the essay in at my office, but do NOT staple them together.

**Journal Homework for class Thursday, January 23:**

- Bring a laptop to class on Thursday! We will use them to start learning how to use a mathematical editing program. Let me know if you have issues bringing a laptop.
- Read the syllabus again. In fact, read it two more times. Although we spent some time on this in class, we did not discuss every detail. You should be sure you have read all of it and understand what is expected. Please ask if you have questions. Note the paper copy I gave you is green so that you can easily find it, but the syllabus is also posted at the top of this website. Refer to it often.
- Check your schedules and requirements for other classes, work, sports, music, etc. Complete the Midterm Exams Survey that I passed out in class on Tuesday. Bring it to class on Thursday. If you need more time to check requirements, ask for an extension. Your answers to the survey are commitments to availability.
- Read the goldenrod laminated strategies (for both problem sets and reading) handout you received today in class. Use this as a constant reference as you work on your homework.
- Before doing the reading, start working through the problems in Section 1.2 (pages 4-5 in the text). In your journal, show your work for problems 2, 4, 6 and 7. You should either copy the question in your journal or make a clear outline of it so that when you look back at your journal to review for an exam you know what the original question was asking. Write a short explanation and/or make a diagram to accompany each answer.
- Read the Preface for Students and Other Learners, and Sections 1.3-1.4 in the text. (This covers pages xxix-xxxvi and 6-14.) This reading assignment is longer than usual because I think reading the Preface is important! Notice how Prof. belcastro states that it is important to start assignments early at the bottom of page xxxvi! She notes that one benefit to this is that it makes you more efficient...i.e. you actually spend less conscious time working on the assignment! (It isn't just me that says this!) She also has great tips for reading mathematics on pages xxxiii-xxxiv, ideas for attacking problems on page xxxv, and tips for writing mathematics on pages xxxvi-xxxvii! You might want to re-read these throughout the semester together with your laminated strategies!
- In your journal, complete the Check Yourself problems at the end of Section 1.3 (page 9) and also at the end of Section 1.4 (page 14). As mentioned above and for the whole semester, you should either copy the question in your journal or make a clear outline of it so that when you look back at your journal to review for an exam you know what the original question was asking; and write a short explanation and/or make a diagram to accompany each answer. (Note that the answers or parts of answers to the Check Yourself problems are at the back of the book...hence the name. Be sure that you work on the problems before you check your answers, and realize that the back does not always give complete solutions.)
- Outline of work for Thursday's journal:
- Section 1.2: Problems 2, 4, 6 and 7
- Notes from reading Section 1.3-1.4, including definitions and statements of principles
- Section 1.3, Check Yourself problems
- Section 1.4, Check Yourself problems
- Your question on the material.

**Collected Homework (Due Friday, January 24 at 3:00pm):**

- The entirety of this first assignment is Collaborative, that is, there is no Individual component.
- Is the sum of an even number and an odd number even or odd? Write a theorem that gives the answer to this question, then prove it. In one sentence, after your proof, describe your strategy for discovering the statement of the theorem. (Note that this question has three parts: stating a theorem, proving the theorem, and stating your strategy.)
- Complete the following problems from Section 1.7 in the textbook (pages 20-21): 6 and 9.
- Notes: The last two problems are counting problems. Be sure to explain your work with sentences and not just show me calculations. The first problem includes a proof. Use the ideas from your reading and we will also discuss this in class on Thursday.
- Tip: It would be good to reread Prof. belcastro's tips for writing mathematics on pages xxxvi-xxxvii before you wrote your final draft of the proof in the first question.